«MAR MAY Jun 6 2016 2017 2018 2 captures 4 Mar 17 - 6 May 17

Close Help

Nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in Banach spaces

Djamila Seba

Received March 14, 2016.   First published January 5, 2017.

Abstract: We consider a nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in a Banach space. The existence of at least one solution is proved by using the set-valued analog of Mönch fixed point theorem associated with the technique of measures of noncompactness.

Keywords: differential inclusion; Caputo fractional derivative; nonlocal boundary conditions; Banach space; existence; fixed point; measure of noncompactness

Classification (MSC 2010): 26A33, 34A60, 34B15

DOI: 10.21136/MB.2017.0041-16

Full text available as PDF.

References:
  [1] R. P. Agarwal, B. Ahmad: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62 (2011), 1200-1214. DOI 10.1016/j.camwa.2011.03.001 | MR 2824708 | Zbl 1228.34009
  [2] R. P. Agarwal, M. Benchohra, D. Seba: On the application of measure of noncompactness to the existence of solutions for fractional differential equations. Result. Math. 55 (2009), 221-230. DOI 10.1007/s00025-009-0434-5 | MR 2571191 | Zbl 1196.2600
  [3] B. Ahmad, A. Alsaedi: Nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions. Bound. Value Probl. (electronic only) (2012), Article ID 124, 10 pages. DOI 10.1186/1687-2770-2012-124 | MR 3017351 | Zbl 1281.34004
  [4] B. Ahmad, J. J. Nieto: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 69 (2008), 3291-3298. DOI 10.1016/j.na.2007.09.018 | MR 2450538 | Zbl 1158.34049
  [5] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, B. N. Sadovskiĭ: Measures of Noncompactness and Condensing Operators. Operator Theory: Advances and Applications 55. Birkhäuser, Basel (1992). DOI 10.1007/978-3-0348-5727-7 | MR 1153247 | Zbl 0748.47045
  [6] A. Alsaedi, S. K. Ntouyas, B. Ahmad: Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions. Abstr. Appl. Anal. 2013 (2013), Article ID 869837, 17 pages. DOI 10.1155/2013/869837 | MR 3049420 | Zbl 1276.26008
  [7] H. H. Alsulami: Application of fixed point theorems for multivalued maps to anti-periodic problems of fractional differential inclusions. Filomat 28 (2014), 91-98. DOI 10.2298/FIL1401091A | MR 3359985
  [8] K. Balachandran, J. Y. Park, J. J. Trujillo: Controllability of nonlinear fractional dynamical systems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 1919-1926. DOI 10.1016/j.na.2011.09.042 | MR 2870885 | Zbl 1277.34006
  [9] J. Banaś, K. Goebel: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics 60. Marcel Dekker, New York (1980). MR 0591679 | Zbl 0441.47056
  [10] M. Benchohra, J. Henderson, D. Seba: Measure of noncompactness and fractional differential equations in Banach spaces. Commun. Appl. Anal. 12 (2008), 419-427. MR 2494987 | Zbl 1182.26007
  [11] M. Benchohra, J. Henderson, D. Seba: Boundary value problems for fractional differential inclusions in Banach spaces. Fract. Differ. Calc. 2 (2012), 99-108. DOI 10.7153/fdc-02-07 | MR 3003005
  [12] M. Benchohra, G. M. N'Guérékata, D. Seba: Measure of noncompactness and nondensely defined semilinear functional differential equations with fractional order. Cubo 12 (2010), 35-48. DOI 10.4067/S0719-06462010000300003 | MR 2779372 | Zbl 1219.34100
  [13] Y. Cui: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51 (2016), 48-54. DOI 10.1016/j.aml.2015.07.002 | MR 3396346 | Zbl 1329.34005
  [14] D. Guo, V. Lakshmikantham, X. Liu: Nonlinear Integral Equations in Abstract Spaces. Mathematics and Its Applications 373. Kluwer Academic Publishers, Dordrecht (1996). DOI 10.1007/978-1-4613-1281-9 | MR 1418859 | Zbl 0866.45004
  [15] J. Han, Y. Liu, J. Zhao: Integral boundary value problems for first order nonlinear impulsive functional integro-differential differential equations. Appl. Math. Comput. 218 (2012), 5002-5009. DOI 10.1016/j.amc.2011.10.067 | MR 2870024 | Zbl 1246.45006
  [16] H.-P. Heinz: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlinear Anal., Theory Methods Appl. 7 (1983), 1351-1371. DOI 10.1016/0362-546X(83)90006-8 | MR 0726478 | Zbl 0528.47046
  [17] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204. Elsevier, Amsterdam (2006). DOI 10.1016/s0304-0208(06)x8001-5 | MR 2218073 | Zbl 1092.45003
  [18] V. Lakshmikantham, S. Leela, J. Vasundhara Devi: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009). Zbl 1188.37002
  [19] A. Lasota, Z. Opial: An application of the Kakutani - Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13 (1965), 781-786. MR 0196178 | Zbl 0151.10703
  [20] H. Mönch: Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal., Theory Methods Appl. 4 (1980), 985-999. DOI 10.1016/0362-546X(80)90010-3 | MR 0586861 | Zbl 0462.34041
  [21] S. K. Ntouyas: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions. Discuss. Math., Differ. Incl. Control Optim. 33 (2013), 17-39. DOI 10.7151/dmdico.1146 | MR 3136580 | Zbl 1307.34016
  [22] S. K. Ntouyas, J. Tariboon: Nonlocal boundary value problems for Langevin fractional differential inclusions with Riemann-Liouville fractional integral boundary conditions. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 22 (2015), 123-141. MR 3360149 | Zbl 1326.34022
  [23] D. O'Regan, R. Precup: Fixed point theorems for set-valued maps and existence principles for integral inclusions. J. Math. Anal. Appl. 245 (2000), 594-612. DOI 10.1006/jmaa.2000.6789 | MR 1758558 | Zbl 0956.47026
  [24] I. Podlubny: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5 (2002), 367-386; Correction: "Geometric and physical interpretation of fractional integration and fractional differentiation", ibid. 6 (2003), 109-110. MR 1967839 | Zbl 1042.26003
  [25] J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado (eds.): Advances in Fractional Calculus-Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007). DOI 10.1007/978-1-4020-6042-7 | MR 2432163 | Zbl 1116.00014
  [26] S. G. Samko, A. A. Kilbas, O. I. Marichev: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). MR 1347689 | Zbl 0818.26003
  [27] S. Szufla: On the application of measure of noncompactness to existence theorems. Rend. Sem. Mat. Univ. Padova 75 (1986), 1-14. MR 0847653 | Zbl 0589.45007
  [28] J. Xu, Z. Wei, W. Dong: Uniqueness of positive solutions for a class of fractional boundary value problems. Appl. Math. Lett. 25 (2012), 590-593. DOI 10.1016/j.aml.2011.09.065 | MR 2856039 | Zbl 1247.34011